Local Computation Algorithms for Graphs of Non-constant Degrees

In the model of local computation algorithms (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on n, the number of vertices. Nonetheless, these complexities are generally at least exponential in d, the upper bound on the degree of the input graph. Instead, we consider the case where parameter d can be moderately dependent on n, and aim for complexities with subexponential dependence on d, while maintaining polylogarithmic dependence on n. We present:a randomized LCA for computing maximal independent sets whose time and space complexities are quasi-polynomial in d and polylogarithmic in n;for constant $$\varepsilon > 0$$ε>0, a randomized LCA that provides a $$(1-\varepsilon )$$(1-ε)-approximation to maximum matching with high probability, whose time and space complexities are polynomial in d and polylogarithmic in n.

[1]  Noga Alon.  A parallel algorithmic version of the local lemma , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[2]  Sofya Raskhodnikova,et al.  Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy , 2013, SIAM J. Comput..

[3]  Ronitt Rubinfeld,et al.  Fast Local Computation Algorithms , 2011, ICS.

[4]  Krzysztof Onak,et al.  A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size , 2011, SODA.

[5]  Bernard Chazelle,et al.  Online geometric reconstruction , 2006, SCG '06.

[6]  Shang-Hua Teng,et al.  A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning , 2008, SIAM J. Comput..

[7]  Dana Ron,et al.  Deterministic Stateless Centralized Local Algorithms for Bounded Degree Graphs , 2014, ESA.

[8]  Sanjeev Khanna,et al.  The Power of Local Information in Social Networks , 2012, WINE.

[9]  Luca Trevisan,et al.  Pseudorandom generators without the XOR Lemma (extended abstract) , 1999, STOC '99.

[10]  Ronitt Rubinfeld,et al.  A Simple Online Competitive Adaptation of Lempel-Ziv Compression with Efficient Random Access Support , 2013, 2013 Data Compression Conference.

[11]  Dana Ron,et al.  Approximating the distance to properties in bounded-degree and general sparse graphs , 2009, TALG.

[12]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[13]  Yishay Mansour,et al.  Converting Online Algorithms to Local Computation Algorithms , 2012, ICALP.

[14]  Krzysztof Onak,et al.  Constant-Time Approximation Algorithms via Local Improvements , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[15]  Michael E. Saks,et al.  Local Monotonicity Reconstruction , 2010, SIAM J. Comput..

[16]  Zeyuan Allen Zhu,et al.  Flow-Based Algorithms for Local Graph Clustering , 2013, SODA.

[17]  Omer Reingold,et al.  New techniques and tighter bounds for local computation algorithms , 2014, J. Comput. Syst. Sci..

[18]  Pierre Kelsen.  Fast parallel matching in expander graphs , 1993, SPAA '93.

[19]  Avinatan Hassidim,et al.  Local computation mechanism design , 2013, EC.

[20]  Ronitt Rubinfeld,et al.  Local Algorithms for Sparse Spanning Graphs , 2014, APPROX-RANDOM.

[21]  Yishay Mansour,et al.  A Local Computation Approximation Scheme to Maximum Matching , 2013, APPROX-RANDOM.

[22]  Zvika Brakerski.  Local Property Restoring , 2008 .

[23]  Klaus Jansen,et al.  Approximation, Randomization and Combinatorial Optimization. , 2008 .

[24]  Ronitt Rubinfeld,et al.  Local Reconstructors and Tolerant Testers for Connectivity and Diameter , 2012, APPROX-RANDOM.

[25]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[26]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[27]  S. Muthukrishnan,et al.  Workload-Optimal Histograms on Streams , 2005, ESA.

[28]  Yuval Peres,et al.  Noise Tolerance of Expanders and Sublinear Expander Reconstruction , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[29]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[30]  Yuichi Yoshida,et al.  Improved Constant-Time Approximation Algorithms for Maximum Matchings and Other Optimization Problems , 2012, SIAM J. Comput..

[31]  Yuichi Yoshida,et al.  An improved constant-time approximation algorithm for maximum~matchings , 2009, STOC '09.

[32]  Fan Chung Graham,et al.  Local Graph Partitioning using PageRank Vectors , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[33]  Boaz Patt-Shamir,et al.  Improved Distributed Approximate Matching , 2015, J. ACM.

[34]  Hsin-Hao Su,et al.  Distributed algorithms for the Lovász local lemma and graph coloring , 2016, Distributed Computing.

[35]  Bernard Chazelle,et al.  Property-Preserving Data Reconstruction , 2004, Algorithmica.

[36]  Luca Trevisan,et al.  Pseudorandom generators without the XOR Lemma , 1999, Electron. Colloquium Comput. Complex..

[37]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[38]  Noga Alon,et al.  Space-efficient local computation algorithms , 2011, SODA.

[39]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[40]  Dana Ron,et al.  Best of Two Local Models: Local Centralized and Local Distributed Algorithms , 2014, ArXiv.

[41]  D. Gamarnik,et al.  Limits of local algorithms over sparse random graphs , 2017 .

[42]  Dana Ron,et al.  On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms , 2007, Electron. Colloquium Comput. Complex..

[43]  Leonid Barenboim,et al.  The Locality of Distributed Symmetry Breaking , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[44]  Boaz Patt-Shamir,et al.  Distributed Approximate Matching , 2009, SIAM J. Comput..

[45]  Dana Ron,et al.  Distributed Maximum Matching in Bounded Degree Graphs , 2015, ICDCN.

[46]  Alon Itai,et al.  A Fast and Simple Randomized Parallel Algorithm for Maximal Matching , 1986, Inf. Process. Lett..